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ABC similar to some permutation of T_aT_bT_c

Source: AIMO 2008, TST 6, P1

January 4, 2009
ratiogeometry unsolvedgeometry

Problem Statement

Let ABC ABC be an acute triangle, and Ma M_a, Mb M_b, Mc M_c be the midpoints of the sides a a, b b, c c. The perpendicular bisectors of a a, b b, c c (passing through Ma M_a, Mb M_b, Mc M_c) intersect the boundary of the triangle again in points Ta T_a, Tb T_b, Tc T_c. Show that if the set of points {A,B,C} \left\{A,B,C\right\} can be mapped to the set {Ta,Tb,Tc} \left\{T_a, T_b, T_c\right\} via a similitude transformation, then two feet of the altitudes of triangle ABC ABC divide the respective triangle sides in the same ratio. (Here, "ratio" means the length of the shorter (or equal) part divided by the length of the longer (or equal) part.) Does the converse statement hold?
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